Chapter 1 Introduction
10.4 Pseudostatic Analysis
These two sets of uniform input were separately applied one component at a time.
In both cases, the strongest response was produced with the stream direction input (channels 12 and 15). This indicates that the stream direction ground motion is the most important component for response to uniform ground motion, so the generated channel 12 record (right abutment) is more severe than the generated channel 15 record (left abutment) when method 1 is used to generate the ground motion. The importance of the stream component for nonuniform input is even further illustrated by the fact that the cross-stream component is obviously larger on the left abutment (channel 17) than the right abutment (channel 14), but the response is larger when 3-component uniform input is supplied from the right abutment records.
One might want to say that the three variations of uniform input ground motion yield responses that are lower and upper bounds for the response to nonuniform in- put. However, the severity of the responses is not the only difference between uniform and nonuniform input. The response of the dam to uniform ground motion has a sig- nificantly different character than nonuniform motion. Generally, for uniform input, the stresses and joint opening are largest in the center of the dam away from the abutments. Cracks open mostly in the center of the dam with very little cracking along the abutments. The major difference between the responses to uniform and nonuniform input is the pseudostatic component of the response. The pseudostatic response is that which would occur if the ground motions are applied very slowly so that inertial and damping effects are negligible. For uniform input, the pseudostatic component is a rigid body motion, but the differential displacements in the nonuni- form input cause pseudostatic deformations of the dam, which are most significant near the abutments.
to 20 meters below the crest. Results of the analysis are shown in Appendix C. The displacement time histories computed at locations corresponding to channels 2–4 are compared from the pseudostatic analysis and the full dynamic analysis in Figure 10.14.
The dynamic analysis does include the pseudostatic component of the response. The dynamic analysis that is compared here uses the model with the water 20 meters below the crest. There is significant dynamic oscillation at channel 2, but the motion at channels 3 and 4 is dominated by the pseudostatic component. Channel 2 is radial at the center of the crest, channel 3 is vertical and channel 4 is tangential at the center of the crest. The time histories indicate that most of the dynamic oscillation of the dam is in the stream direction.
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0 1 2 3 4 5 6 7 8 9 10 11
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Time (sec)
Figure 10.14: Displacement time histories at locations corresponding to channels 2–4 computed from a nonlinear pseudostatic analysis compared to the time histories from a nonlinear dynamic analysis (method 1)
The maximum compressive stresses in the arch direction are shown in Figure 10.15.
On the upstream face, the pseudostatic stresses are large along the upper abutments, particularly the left abutment. On the downstream face, the pseudostatic stresses are largest near the upper left abutment, but there are also significant compressive stresses in the entire half of the dam that is closer to the left abutment. The stresses
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Figure 10.15: Maximum compressive arch stresses (MPa) computed during a nonlin- ear pseudostatic analysis with nonuniform ground motion input (method 1)
for the dynamic analysis along the abutments and in the center of the downstream face are dominated by these pseudostatic stresses, and the dynamic effects are important near the center of the crest. The maximum joint opening throughout the dam has a similar distribution to the arch compression on the upstream face. Like the stresses, the joint opening along the abutments is pseudostatic and the joint opening in the interior of the dam is caused by oscillation of the dam. Cracking is minimal for the pseudostatic analysis. Only three elements along the abutments crack and the opening is small.
The high stresses at the upper left abutment are not the result of a stress con- centration that may be present, because the singularity would also cause similar high stresses for uniform ground motion input. However, that is not the case so any stress concentration is too weak to show an effect. The large pseudostatic stresses at the upper left abutment appear to be related to the large cross-stream displacement in the generated input along the left abutment. The largest amplification of displacement in the generated ground motions is the cross-stream component along the left abut-
ment. Channel 17 illustrates this large displacement. The high stresses do not arise from an overall compression of the arch of the dam, but from the local differential in displacement input as the elevation increases along the abutment. This is shown by prescribing the ground motions generated for the left abutment to both abutments at the same time, so that the arch is not compressed along the crest. The pseudo- static analysis with this input shows large arch compression at both abutments (see Figure 10.16). Thus, the large input cross-stream displacement causes large stresses at each abutment at the times when the input motion locally compresses each side of the dam. This effect is the reason that nonuniform ground motions generated by methods 5 and 9 induce larger maximum arch compression at the upper left abutment than ground motion from method 1. The generated channel 17 displacement is larger in the direction causing compression of the dam for methods 5 and 9 than it is for method 1.
While the pseudostatic component of the response does not cause significant crack-
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Figure 10.16: Maximum compressive arch stresses (MPa) computed during a nonlin- ear pseudostatic analysis with nonuniform ground motion input (method 1) with the left abutment input also prescribed on the right abutment
ing, the pseudostatic stresses and joint opening are an important part of the re- sponse for nonuniform ground motion. The pseudostatic component of the response to nonuniform ground motion has been observed to be important by other researchers (Lin et al., 1996; Mojtahedi and Fenves, 2000), but in another study with nonuniform ground motion it was not as important (Nowak and Hall, 1990). Therefore, the degree of importance of the pseudostatic response can depend significantly on the nonuni- form seismic input that is used, and care should be taken in choosing the input. The displacements, in particular, need to be carefully integrated from the accelerations.
Recall that the method 1 generated ground motions for the 2001 earthquake agree best in terms of displacements when being compared to the actual records. This is good since accurately capturing the pseudostatic response is important. The ap- proach used here to generate ground motions shows promise based on the available data, but more data and more analysis is necessary for validation.
Chapter 11
Summary and Conclusions
The occurrence of a relatively small earthquake (magnitude 4.3) on January 13, 2001, at Pacoima Dam presented an opportunity to study the spatial nonuniformity of the ground motion along the abutments of an arch dam. The ground motion is amplified by the topography of the canyon in which the dam is situated, and assuming that the seismic waves propagate upward, the ground motion arrives at the top of the dam later than at the base. The spatial nonuniformity was captured by an array of accelerometers located at Pacoima Dam that includes 3-component measurements at three locations along the canyon.
In studying the ground motion, a system identification study was done with MODE-ID using the acceleration records from the January 2001 earthquake. The estimates from MODE-ID indicate that the response was dominated by the first two modes. The modes are generally symmetric (4.73 Hz–4.83 Hz) and antisymmetric (5.06 Hz), but Pacoima Dam is not a perfectly symmetric structure so the modes are not perfectly symmetric and antisymmetric. The estimated modal damping is around 6% to 7% of critical for both modes. It was initially believed that Pacoima Dam responded elastically to the 2001 earthquake. However, the natural frequencies identified by MODE-ID are significantly lower than the frequencies for the symmetric and antisymmetric modes determined from forced vibration tests performed in 1980 (5.45 Hz and 5.60 Hz). The mode shapes are somewhat similar, but the decrease in frequencies is substantial for both modes. The significant stiffness reduction shows that either the system has been damaged since 1980 and left unrepaired, which is not
the case, or the 2001 earthquake induced a nonlinear response from Pacoima Dam.
When short windows of the 2001 records are provided to MODE-ID, the natural fre- quencies are shown to vary during the course of the earthquake. The frequencies tend to increase as the motion from the earthquake dies out, but the frequencies do not reach the values determined from the 1980 forced vibration tests.
Partial records were also recorded by the accelerometer array at Pacoima Dam during the 1994 Northridge earthquake, and these records were also provided in short windows to MODE-ID. The results identify a system that has significant nonlinear- ity. At the beginning of the earthquake, the system is similar to the system identified from the 2001 earthquake, but as the Northridge earthquake progresses the natu- ral frequencies decrease significantly, indicating a reduction of stiffness. This makes sense because the Northridge earthquake was observed to cause substantial damage to Pacoima Dam and its foundation. However, no damage was reported after the 2001 earthquake and the motion was not believed to be large enough for the dam to leave the elastic range.
In order to investigate the apparent nonlinear response during the 2001 earth- quake, a forced vibration experiment was performed in July and August 2002. These tests yielded modal frequencies and mode shapes that were similar to the 1980 find- ings. The symmetric mode and antisymmetric mode frequencies were determined to be bounded by 5.35 Hz–5.45 Hz and 5.65 Hz–5.75 Hz, respectively, compared to 5.45 Hz and 5.60 Hz in 1980. The modal damping determined from the 2002 ex- periment ranges from 4% to 7% for the symmetric mode and 4.5% to 5.5% for the antisymmetric mode. The damping computed from the 1980 tests is higher, but it is believed to be overestimated based on the quality of the data. The reservoir was actually 13 meters deeper in 1980 so the natural frequencies identified in 1980 should be lower due to the added mass of the water. The fact that the symmetric mode frequency was lower in 2002 indicates that the Pacoima Dam system may have lost stiffness in the 22 years between experiments. This most likely happened in 1994 when the dam and foundation were damaged by the Northridge earthquake. Repairs after the earthquake may not have returned the dam to its pre-earthquake state, but
the difference does not explain the much larger decrease in stiffness that the system identification with the 2001 records seems to indicate. The differences are far too large to explain by the fact that the modes are closely spaced. The frequencies are believed to be good estimates from both MODE-ID and forced vibration. The slightly lower damping from the forced vibration tests could possibly be a result of closely spaced modes that make accurate damping estimates difficult, but there must have been some form of nonlinearity during the 2001 earthquake to explain the stiffness changes.
The nonlinearity in the 2001 earthquake response is believed to be in the foun- dation rock, particularly at the upper left abutment. The rock at the upper left abutment was fractured after both the 1971 San Fernando earthquake and the 1994 Northridge earthquake, and in both cases repairs were made. The foundation must remain in a state in which stiffness can be lost during even low level excitation, but the foundation does not permanently lose stiffness if no permanent displacements of the rock are caused. The nonlinear behavior must not be engaged by the forced vibration tests because that excitation is orders of magnitude smaller than the 2001 earthquake. Also, while seismic waves travel to the dam through the foundation, forced vibration excitation originates on the crest of the dam. Thus, the earthquake excitation may more easily affect the foundation. The dam concrete is believed to have behaved elastically during the 2001 earthquake.
This nonlinear effect may be even more significant than initially thought. The MODE-ID identified system is a hybrid of the systems with flexible and rigid foun- dations, so this system should actually be stiffer than the system with a flexible foundation. The cross-correlation functions of ambient measurements can actually be shown to oscillate at the natural frequencies of the flexible foundation system, and cross-correlations of earthquake records may be a close enough approximation to have the same properties. When cross-correlation functions for the 2001 records are provided as free vibration output for MODE-ID, the symmetric mode frequency is much lower than the estimate using the records directly. However, the antisymmetric mode frequency is actually found to be higher through the cross-correlation functions.
Also, the computed response from a finite element model that is calibrated to match the MODE-ID estimates from the records seems to indicate that the symmetric mode frequency is too high and that the antisymmetric mode frequency is too low to match the actual recorded response. These results are not completely intuitive because the antisymmetric mode frequency is not expected to be higher for what seems to be a more flexible system. While the specific nature of the system that is identified by MODE-ID is not clear, it is apparent that the dam oscillated as a stiffer system during the forced vibration tests than it did during the January 2001 earthquake.
The variation of stiffness has implications for structural health monitoring. The state of a structure can be monitored by tracking the modal properties of the sys- tem. In the case of Pacoima Dam, the January 2001 earthquake was believed to be small enough to induce a linear response, which theoretically should be able to be compared to forced vibration tests to determine whether the state of the dam has changed between the two events. However, this was shown not to be the case, be- cause while the 2002 forced vibration tests indicate a different state than the 2001 earthquake, there were no significant changes to the dam system between the events.
If the earthquake response was compared to the 1980 forced vibration tests without knowledge of the 2002 tests, the frequency variations could have been interpreted as a permanent stiffness reduction that did not actually happen. The same problem may exist when comparing forced vibration data to ambient data. Perhaps, nonlinear response is significant even for these excitations. It is believed that structural health monitoring should be useful as long as responses from the same type of excitation are compared, but even this may not be true for all structures. The precise level of the excitation may even be important. Comparisons for structural monitoring need to be made before and after an event based on ambient vibration or forced vibration. This is not a significant limitation, especially if ambient data can be collected in real-time.
Comparisons from different excitations are not reliable, at least for Pacoima Dam, and more study is necessary to fully understand the nonlinear mechanism in Pacoima Dam.
A main goal of this research is to develop a useful method for generating nonuni-
form ground motion to be used in structural analyses of dams. The SCADA finite element program was modified to accept nonuniform input along the abutments of the dam. A preprocessor was also created to interpolate ground motion to each node of a finite element model from a nonuniform set of ground motion records that include at least three locations along the abutments: one near the base of the dam and one each at higher elevations along the right and left abutments. The interpolation is done in the frequency domain so that amplitude and phase can both be interpolated. The input to the model is assumed to be free-field. This is not the reality when actual recordings from a dam are used, but the approximation is necessary since no free-field data is available.
The finite element model that was constructed has 110 dam elements (6 of which model the thrust block), 1320 water elements and 728 foundation elements. The dam elements can include contact nonlinearities through the smeared crack method. The water and foundation are modeled linearly with water compressibility neglected and mass omitted from the foundation. This model can be reasonably calibrated to match the results from the 2002 forced vibration experiment, and softened to calibrate to the modal estimates from the 2001 earthquake records. The model can be subjected to temperature cycles, and if joints are omitted from the dam the computed response at the crest of the dam agrees well with annual cycles of GPS and temperature data that were analyzed in the late 1990s. This indicates that the contraction joints in Pacoima Dam are closed throughout the year. Also, damage that was sustained during the 1971 San Fernando earthquake can be simulated with the model by softening a section of the foundation and disconnecting three nodes between the dam and the thrust block.
The modal properties of the damaged model reasonably match results from forced vibration tests done in 1971 before repairs were made.
The SCADA analysis can be run in a linear mode or a nonlinear mode. The analysis with the January 2001 earthquake ground motion indicates that the dam probably did vibrate linearly, but no conclusion can be made about the foundation.
The modeled response compares well to the recorded motion, but the computed ac- celerations on the dam body overestimate the records during the strongest motion.
The computed displacements agree with the records better than the accelerations.
The reason for the overestimation is not known. Perhaps, the modal damping should be higher, but the damping was based on system identification that was done with the 2001 earthquake records so there is no basis for increasing the damping. The agreement of the modeled response with the recorded response is good enough to use for assessing the method that is proposed for generating nonuniform ground motion.
The approach for generating ground motion is based on the recordings from the abutments on January 13, 2001. The topographic amplification is characterized by transfer functions comparing the records on the abutments to the records at the base of the dam. The amplification is frequency-dependent. Spectral displacement ratios, both 0% and 5% damped, and Fourier amplitude spectra are considered. For the generation method, the amplification would, ideally, be implemented as an average amplification determined from several events. In order to be consistent with this, a piecewise linear approximation to the spectral displacement ratios is also considered.
The time for the seismic waves to travel from the base of the dam upward along the abutments is considered through time delays based on cross-correlations and the ac- tual relative phase between records from the Fourier spectra. In order to characterize the ground motion with time delays, the importance of reflections of the traveling waves is assumed to be small, which appears to be reasonable. Frequency-dependent delays are computed by taking the time at which the maximum value occurs from the cross-correlation of the displacement responses of 5% damped single degree of freedom oscillators at each frequency excited by the acceleration records. Frequency- independent delays are computed by taking the time at which the maximum value occurs from the cross-correlation of the acceleration records. The delays are converted to relative phase for generating ground motions. The various amplification functions and relative phase functions are used to generate motions at points along the abut- ments from a reference 3-component ground motion at the base of the dam. This is done in the frequency domain, and then the ground motions are converted back to the time domain for use in dynamic analyses.
Nonuniform ground motions were generated from the base records of the Jan-